ATOMIC SPECTROSCOPY AND OPACITY

Transcription

1 ATOMIC SPECTROSCOPY AND OPACITY 3 Weeks Lecture Course Textbook: Atomic Astrophysics and Spectroscopy -A.K. Pradhan and S.N. Nahar (Cambridge, 2011) SULTANA N. NAHAR Department of Astronomy The Ohio State University Columbus, Ohio, USA naharastronomy.ohio-state.edu Web: nahar Physics Department Cairo University, Giza, Egypt February 16 - March 7, 2013 Support: DOE, NSF, OSC, Cairo University 1

2 ATOMIC SPECTROSCOPY AND OPACITY 3 Weeks Lecture Course Textbook: Atomic Astrophysics and Spectroscopy (AAS) -A.K. Pradhan and S.N. Nahar (Cambridge, 2011) Lecture Topics: Week 1: Physics and Astronomy Connection i) Elements and Spectroscopy ii) Opacity and Observations iii) The Sun and Evolution of Spectroscopy iv) Electron and Photon Distribution Functions v) The Plasma Universe Atomic Structure and Spectroscopy i) Hydrogen - Single electron energy and wave functions ii) Quantum numbers n,l,m, s, parity iii) Spectral lines - Rydberg formula Week 2: Atomic Structure, Computation with SUPERSTRUCTURE & Atomic Processes i) Multi-election Systems - Angular Sates ii) Angular Couplings, Quantum Defects iii) Hartee-Fock Approximation iv) Central Field Approximations v) Dirac Equation, Breit-Pauli Approximation, Dirac-Fock 2

3 Approximation vi) Atomic Processes of Plasma Opacity Week 3: Plasma Opacity, R-matrix Calculations for Atomic Processes i) Autoionization and Resonances and Opacity ii) Plasma Opacity iii) Close-Coupling Approximation and R-matrix Method iv) R-matrix Calculations 3

4 i) Elements and Spectroscopy: ELEMENTS (Key Information) Periodic Table of elements (Ref. NIST) Period Group 1 IA 1 2 S 1/2 H Hydrogen s S 1/2 Li Lithium s 2 2s Na Sodium 2 S 1/ [Ne]3s K 2 S 1/2 Potassium [Ar]4s Rb Rubidium [Kr]5s Cs Cesium [Xe]6s Fr Francium (223) [Rn]7s IIA 4 Be 1 S 0 Beryllium s 2 2s Mg Magnesium [Ne]3s S 0 20 Ca Calcium S 0 [Ar]4s Sr Strontium S 1/2 1 S 0 [Kr]5s Ba Barium S 1/2 1 S 0 [Xe]6s Ra Radium (226) 2 S 1/2 1 S 0 [Rn]7s P E R I O D I C T A B L E Atomic Properties of the Elements 3 IIIB 21 Sc Scandium [Ar]3d4s 2 [Ar]3d 2 4s Y Yttrium Frequently used fundamental physical constants For the most accurate values of these and other constants, visit physics.nist.gov/constants 1 second = periods of radiation corresponding to the transition between the two hyperfine levels of the ground state of 133 Cs speed of light in vacuum c m s 1 (exact) Planck constant h J s ( /2 ) elementary charge electron mass proton mass fine-structure constant e m e m e c 2 m p C kg MeV kg 1/ Rydberg constant R m 1 R c R hc Hz ev Boltzmann constant k J K 1 4 IVB 22 Ti Titanium D 3/2 3 F 2 40 Zr Zirconium D 3/2 3 F 2 [Kr]4d5s 2 [Kr]4d 2 5s Hf 3 F 2 Hafnium Nb Niobium [Kr]4d 4 5s Mo Molybdenum [Kr]4d 5 5s W Tungsten Tc Technetium (98) [Kr]4d 5 5s Re 6 S 5/2 Rhenium Ru Ruthenium [Kr]4d 7 5s Os 5 D 4 Osmium Rh 4 F 9/2 Rhodium [Kr]4d 8 5s Ir 4 F 9/2 Iridium Solids Liquids Gases Artificially Prepared 29 Cu Copper [Xe]4f 14 5d 2 6s 2 [Xe]4f 14 5d 3 6s 2 [Xe]4f 14 5d 4 6s 2 [Xe]4f 14 5d 5 6s 2 [Xe]4f 14 5d 6 6s 2 [Xe]4f 14 5d 7 6s 2 [Xe]4f 14 5d 9 6s [Xe]4f 14 5d 10 6s [Xe]4f 14 5d 10 6s Al Si P S Cl Aluminum Silicon Phosphorus Sulfur Chlorine Ar Argon [Ne]3s 2 3p [Ne]3s 2 3p 2 [Ne]3s 2 3p 3 [Ne]3s 2 3p 4 [Ne]3s 2 3p 5 [Ne]3s 2 3p [Ar]3d 8 4s 2 [Ar]3d 10 4s [Ar]3d 10 4s 2 [Ar]3d 10 4s 2 4p [Ar]3d 10 4s 2 4p 2 [Ar]3d 10 4s 2 4p 3 [Ar]3d 10 4s 2 4p 4 [Ar]3d 10 4s 2 4p 5 [Ar]3d 10 4s 2 4p S 0 2 S 1/2 46 Pd Palladium [Kr]4d Pt 3 D 3 Platinum Ag Silver [Kr]4d 10 5s Au Gold 48 Cd Cadmium [Kr]4d 10 5s Hg Mercury F? Rf Rutherfordium (261) [Rn]5f 14 6d 2 7s 2? 6.0? 5 VB 23 V Vanadium [Ar]3d 3 4s Ta Tantalum Db Dubnium (262) 6 VIB 24 Cr Chromium F 3/2 7 S 3 [Ar]3d 5 4s D 1/2 7 S 3 4 F 3/2 5 D 0 Sg Seaborgium (266) 7 VIIB 25 Mn Manganese [Ar]3d 5 4s 2 [Ar]3d 6 4s Bh Bohrium (264) 26 Fe Iron S 5/2 5 D 4 6 S 5/ VIII IB 5 F 5 Hs Hassium (277) 27 Co Cobalt 4 F 9/ [Ar]3d 7 4s Mt Meitnerium (268) 28 Ni Nickel Uun Ununnilium (281) 3 F 4 2 S 1/2 2 S 1/2 Uuu Unununium (272) 12 IIB 30 Zn Zinc Uub Ununbium (285) IIIA IVA VA VIA VIIA C N O F Carbon Nitrogen Oxygen Fluorine B Boron Ga Gallium S 0 2 P 1/2 1 S 0 1 S 0 Physics Laboratory physics.nist.gov 49 In Indium 50 Sn Tin Sb Antimony Te Tellurium I Iodine [Kr]4d 10 5s 2 5p [Kr]4d 10 5s 2 5p 2 [Kr]4d 10 5s 2 5p 3 [Kr]4d 10 5s 2 5p 4 [Kr]4d 10 5s 2 5p 5 [Kr]4d 10 5s 2 5p Tl 2 P 1/2 1s 2 2s 2 2p Thallium [Hg]6p Ge Germanium Pb Lead P 0 1s 2 2s 2 2p As Arsenic 83 Bi Bismuth Po Polonium (209) 85 At Astatine (210) 86 Rn Radon (222) [Hg]6p 2 [Hg]6p 3 [Hg]6p 4 [Hg]6p 5 [Hg]6p Uuq Ununquadium (289) Standard Reference Data Group 4 S 3/2 1s 2 2s 2 2p P 2 1s 2 2s 2 2p 4 34 Se Selenium Uuh Ununhexium (292) 2 P 3/2 1s 2 2s 2 2p Br Bromine VIIIA 2 He Helium s Ne Neon S 0 1 S 0 1s 2 2s 2 2p 6 2 P 1/2 3 P 0 4 S 3/2 3 P 2 2 P 3/2 1 S 0 36 Kr Krypton P 0 4 S 3/2 3 P 2 2 P 3/2 1 S 0 54 Xe Xenon P 1/2 3 P 0 4 S 3/2 3 P 2 2 P 3/2 1 S 0 2 P 1/2 3 P 0 4 S 3/2 3 P 2 2 P 3/2 1 S 0 Symbol Name Atomic Weight Atomic Number Ground-state Configuration Ground-state Level 58 Ce Cerium 1 G [Xe]4f5d6s Ionization Energy (ev) Lanthanides Actinides 57 La Lanthanum [Xe]5d6s Ac Actinium (227) [Rn]6d7s Ce Cerium D 3/2 1 G 4 [Xe]4f5d6s Th Thorium D 3/2 3 F 2 [Rn]6d 2 7s Pr 4 I Nd 5 9/2 I Pm 6 4 H Sm 7 Eu Gd Tb Dy 5 I Ho 4 5/2 F 8 0 S 9 7/2 D 6 2 H 8 I Er 3 15/2 15/2 H Tm 2 6 F 7/2 Praseodymium Neodymium Promethium Samarium Europium Gadolinium Terbium Dysprosium Holmium Erbium Thulium (145) [Xe]4f 3 6s Pa Protactinium [Rn]5f 2 6d7s K 11/2 [Xe]4f 4 6s L 6 92 U Uranium [Rn]5f 3 6d7s [Xe]4f 5 6s Np 6 L 11/2 Neptunium (237) [Rn]5f 4 6d7s [Xe]4f 6 6s 2 [Xe]4f 7 6s 2 [Xe]4f 7 5d6s 2 [Xe]4f 9 6s 2 [Xe]4f 10 6s 2 [Xe]4f 11 6s 2 [Xe]4f 12 6s 2 [Xe]4f 13 6s Pu F Am Cm D Bk Cf I Es I Fm Md 0 S 7/2 2 H 15/2 8 15/2 H 6 F 7/2 Plutonium Americium Curium Berkelium Fermium Mendelevium (244) (243) (247) (247) (257) (258) [Rn]5f 12 7s 2 [Rn]5f 13 7s 2 [Rn]5f 6 7s [Rn]5f 7 7s [Rn]5f 7 6d7s [Rn]5f 9 7s Californium (251) [Rn]5f 10 7s Einsteinium (252) [Rn]5f 11 7s Yb 1 S 0 Ytterbium [Xe]4f 14 6s No 1 S 0 Nobelium (259) [Rn]5f 14 7s Lu 2 D 3/2 Lutetium [Xe]4f 14 5d6s P 1/2? 103 Lr Lawrencium (262) [Rn]5f 14 7s 2 7p? 4.9? Based upon 12 C. () indicates the mass number of the most stable isotope. For a description of the data, visit physics.nist.gov/data NIST SP 966 (September 2003) Elements: Gases (pink), Solids (white),i Liquids (blu Artificially Prepared (yellow) 4

5 Abundant Elements: H (X - 90%, mass:70%), He (Y - 7%, mass:28%) Other elements (metals) (Z- 3%, mass: 2%): Li, Be, B, C, N, O, F, Ne, Na, Mg, Al, Si, S, Ar, Ca, Ti, Cr, Fe, Ni - Fusion cycle in a star usually ends at Fe (strongest nuclear force) Study elements through spectroscopy Although constitute only 2% of the plasma, the metals are responsible for most of the spectral features - they crucially determine properties such as the plasma opacity that governs the transfer of radiation through the source 5

6 Heavier elements - Supernova explosion Solar system made from debris of supernova explosions TYCHO SUPERNOVA REMNANT IN CASSIOPI Low Resolution Observation: Spitzer (red - Infrared - dust), Calar Alto (O, white stars), Chandra (X-rays - yellow, green - Expanding debris, blue - Ultra-energetic electrons) 6

7 SOLAR ABUNDANCES Solar abundances (A k, usually in log scale): H - 90 % (by number), 70% (by mass fraction) He - 10 % (by number), 28 % (by mass) Metals - 2% (by mass). O (highest), C, N, Ne, Mg, Si, S, Ar, Fe Traditionally H abundance: log(a H ) = 12, other elements are scaled relative to it Table: A mixture of standard solar abundances (10% uncertainties, Seaton et al. 1994) Element (k) Log A k A k /A H H He (-1) C (-4) N (-5) O (-4) Ne (-4) Na (-6) Mg (-5) Al (-6) Si (-5) S (-5) Ar (-6) Ca (-6) Cr (-7) Mn (-7) Fe (-5) Ni (-6) 7

8 A supernova remnant: an outer shimmering shell of expelled material, and a core skeleton of a once-massive star Heavy elements, beyond Iron, are generated from nuclear fusion during supernova explosions and are scattered into interstellar medium Two pure H & He clouds, Clouds, detected first in 2010 Pristine End of a star (98%): Below 1.4 solar mass (Chandra limit): Core becomes a white dwarf - diomond formation (electron repulsion equals gravity) Above 1.4 but below 4 solar mass: neutron star (electrons pressed to Neutron formation - largher gravity) Larger mass - blackhole (extreme gravity) 8

9 White Dwarf stars - cosmic clocks. The age of the universe is at least as big as the oldest stars in it. This principle can map the ages of galactic compoments 9

10 ii) Opacity and Observation STUDYING ASTRONOMICAL OBJECTS ASTRONOMICAL objects are studied in three ways: Imaging: - Beautiful pictures or images of astronomical objects, Stars, Nebulae, Active Galactic Nuclei (AGN), Blackhole Environments, etc Provides information of size and location of the objects Photometry: Bands of Electromagnetic Colors ranging from X-ray to Radio waves macroscopic information and low resolution spectroscopy Spectroscopy: Taken by spectrometer - Provides most of the detailed knowledge: temperature, density, extent, chemical composition, etc. of astronomical objects Spectroscopy is underpinned by Atomic & Molecular Physics 10

11 OPACITY Opacity is a measure of radiation transfer Higher opacity - less radiation and lower opacity - more radiation transfer 11

12 ATMOSPHERIC OPACITY ( Higher opacity - less radiation and lower opacity - more radiation reaching earths surface Opacity determines types of telescopes - earth based or space based Gamma, X-ray, UV are blocked while visible light passes through CO 2, H 2 O vapor, other gases absorb most of the infrared frequencies Part of radio frequencies is absorbed by H 2 O and O 2, and part passes through 12

13 iii) The Sun and Evolution of Spectroscopy Light is a mixture of colors - Spectrum: splitting of colors For example: Rainbow is the spectrum of white sunlight Each atom gives out light with its own set of colors Spectrum is lines of colors, like the barcode on an item The lines key information of astronomical objects the blackhole or other astrophysical objects Spectrum of Carbon 13

14 Solar Spectra: Absorption & Emission Lines Absorption line - by absorption of a photon as an electon jumps to a higher level Emission line - by emission of a photon as electron dumps to a lower level An absorption or emission line from the same two energy levels appear at the same energy position in the spectrum Fraunhofer lines (1815) in the Solar Spectrum Used alphabet - no element assignment; Later spectoscopy with quantum mechanics: A (7594 Å,O), B (6867 Å,O) (air), C (6563 Å H), D1 & D2 (5896, 5890 Å Na, yellow sun), E(Fe I), G(CH), H $ K (Ca II) [(Courtesy: Institute for Astronomy, University of Hawaii] 14

15 SUNLIGHT ON THE EARTH SURFACE ( Spectrum png) The Sun - Blackbody (constant T), gives out radiation in Planck distribution Solar surface T = 5770 K peak blackbody emission - yellow 5500 Å. H 2 O has inefficient absorption windows: 3 are molecular IR bands - J, H, and K bands Solar UV flux drops off rapidly by ozone effect preventing it to reach earth 15

16 iv) Electron and Photon Distribution Functions A plasma of charged particles and a radiation field of photons treated with two distribution functions, Planck and Maxwell Temperature is defined through a radiation (photon) or the particle (electron) energy distribution kinetic T defined from E = hν kt, or E = 1/2mv 2 = 3/2kT Consider a star ionizing a molecular cloud into a gaseous nebula - The two objects, the star & the nebula, have different T distribution functions Energy of the radiation emitted by the star - Planck distribution function Energies of electrons in the surrounding ionized gas heated by the star - Maxwell distribution 16

17 PLANCK DISTRIBUTION FUNCTION total energy emitted by an object s related to its T by the Stefan-Boltzmann Law E = σt 4 (1) σ = Watts/(m 2 K 4 )= Stefan constant Radiation of a star, a blackbody, is given by the Planck distribution function B ν (T ) = 2hν3 1 c 2 exp(hν/kt ) 1, (2) T =radiation temperature, ν = frequency of the photons. MAXWELL DISTRIBUTION FUNCTION The charged particles in the plasma ionized by a star in an H II region has an electron temperature T e associated with the mean kinetic energy of the electrons kt e 1 2 mv2 = 3 2 kt e. (3) 17

18 The velocity (or energy) distribution of electrons is characterized by a Maxwellian function at temperature T e, f(v) = 4 ( m ) 3/2 v 2 exp π 2kT ( mv2 2kT ). (4) 18

19 PLANCK & MAXWELL DISTRIBUTIONS B λ (T * ) 7000 [K] Planck distribution 5770 [K] 4000 [K] λ [A] f(e,t e ) (K) Maxwellian distribution (K) (K) E (ev) H II region (nebula): Ionized by blackbody radiation of T K, electron kinetic energy of Maxwellian distribution at T e K 19

20 v) The Plasma Universe PLASMA COVERS VAST REGION (99%) IN T-ρ PHASE SPACE (AAS, Pradhan & Nahar, 2011) 9 8 tokamaks ICF log[ T (K) ] nebulae stellar coronae stellar interiors BLR AGN stellar atmospheres Jovian interior log[ N e (cm 3 )] Temperature-density regimes of plasmas in astrophysical objects - BLR-AGN ( broad-line regions) in active galactic nuclei, where many spectral features associated with the central massive black hole activity manifest themselves Laboratory plasmas - tokamaks (magnetic confinement fusion devices), Z-pinch machines (inertial confinement fusion (ICF) devices) 20

21 WARM & HOT DENSE MATTER (HEDLP-FESAC report) Hot Dense Matter (HDM): - Sun s ρ-t track, Supernovae, Stellar Interiors, Accretion Disks, Blackhole environments - Laboratory plasmas in fusion devices: inertial confinement - laser produced (NIF) and Z pinches (e.g. Sandia), magnetic confinement (tokamaks) Warm Dense Matter (WDM): - cores of large gaseous planets 21

22 X-RAYS FROM A BLACKHOLE - CENTAURUS A GALAXY (Observed by Chandra space telescope) Photometric image: red - low-energy X-rays, green - intermediate-energy X-rays, and blue - the highest-energy X-rays. The dark green and blue bands are dust lanes that absorb X-rays. Highly energetic SUPERHOT ATOMS encircling the blackhole are in a plasma state & emit bright K α X-rays Electrons are ripped off - highly charged state, e.g. He- & Li-like Ne,SI, S, Fe; - emit mainly K α (1s - 2p) photons Sucked in materials are ejected as a jet (L & E conservation) Blasting from the black hole a jet of a billion solar-masses extending to 13,000 light years 22

23 SPECTRUM of the Wind near Blackhole: GRO J Binary Star System Materials from the large star is sucked into the companion blackhole - form wind as they spiral to it (Miller et al., 2006) Spectrum: Highly charged Mg, Si, Fe, Ni lines Red Spectrum - Elements in natural widths Doppler Blue Shift - Wind is blowing toward us 23

24 Signature of a Blackhole The well-known K α (1s-2p) transition array lines of iron near a black hole in Seyfert I galaxy MCG (ASCA & Chandra obs) The maximum energy for a 1s-2p transition in iron is 6.4 kev. However, the large extension of the lines toward low energy means that the escaped photons have lost energies in the black hole. Gravitational broadening of Fe K α emission line from close vicinity of the black hole. 24

25 SOLAR ABUNDANCES & HED PLASMA Sun s interior - nuclear core to the end of convection zone beyond which the radiation escapes At the convection zone boundary, R CZ, the temperature T e 193 ev, density n e /cm 3 (HED - High Energy Density - Fe ions (Fe XVII-XIX)) Takes over a million years for radiation created in the core (gamma rays) to travel to surface: Reason - OPACITY Although the most studied star, we can not explain all observations. Recent determination of solar abundances, from measurements and 3D hydro NLTE models, show 30-40% lower abundances of C, N, O, Ne, Ar than the standard abundances; contradict the accurate helioseismology data 25

26 CREATION OF HED PLASMA: NIF, LLNL NIF: Pulses from 192 high-power lasers arrive (in s of each other) (TOP, spike - target position) to center target (mm-size) MJ shot (Bottom) 26

27 HED PLASMA, Z PINCH, Sandia Natl lab Magnetically driven z-pinch implosions convert electrical energy into radiation. Internal shock heating X-rays - Energy: 1.5 MJ, Power: 200 TW (Latest: 27 Mamp) Te 190 ev, density ne /cm3, similar to that at solar (RRZ ) 27

28 DISCREPANCY BETWEEN EXPERIMENTAL & THEORETICAL OPACITIES (Bailey, Sandia lab) Z PINCH spectra, SNL: Plasma - T e 193 ev, Density n e /cm 3 - similar at solar R CZ Observed (red), Calculated (blue). Top: Diagnostic lines, Bottom: Iron - Large differences Reason: OPACITY Serious discrepancies for Fe opacity Experimental n e is wrong OR bound- free absorption (photoionization) is inaccurate 28

29 ATOMIC PROCESSES FOR ASTROPHYSICAL SPECTROSCOPY 1. Photo-Excitation & De-excitation: X +Z + hν X +Z An Electron jumps up or down to level, but does not leave the atom Fig:Top- Energy levels C I, C II Energy Levels Configuration 2S+1 L π 2s2p 2 2 P 2 S 2 D 4 P C II: 2s 2 2p 2 P o 2s2p 3 C I: 2s 2 2p 2 1 P o 3 P o 1 D o 3 D o 3 S o 5 S o 1 S 1 D 3 P 29

30 Photo-Excitation and Opacity A 21 - Spontaneous Decay B 21 - Stimulated Decay with a radiation field Intoduces absorption or emission lines Atomic quantities (constant numbers): - Oscillator Strength (f) - Radiative Decay Rate (A-value) Monochromatic opacity (κ ν ) depends on f ij κν(i j) = πe2 mc N if ij φ ν (5) N i = ion density in state i, φ ν = profile factor κ includes thousands to millions of transitions 30

31 ALLOWED & FORBIDDEN TRANSITION Determined by angular momentum selection rules i) Allowed: Electric Dipole (E1) transitions ( J = 0,±1, L = 0,±1,±2; parity changes, same-spin & different spin or intercombination transition) Forbidden: ii) Electric quadrupole (E2) transitions ( J = 0,±1,±2, parity does not change) iii) Magnetic dipole (M1) transitions ( J = 0,±1, parity does not change) iv) Electric octupole (E3) transitions ( J= ±2, ±3, parity changes) v) Magnetic quadrupole (M2) transitions ( J = ±2, parity changes) Allowed transitions >> Forbidden transitions 31

32 EX: ALLOWED & FORBIDDEN TRANSITIONS Diagnostic Lines of He-like Ions: w,x,y,z w(e1) : 1s2p( 1 P o 1) 1s 2 ( 1 S 0 ) (Allowed Resonant) x(m2) : 1s2p( 3 P o 2) 1s 2 ( 1 S 0 ) (Forbidden) y(e1) : 1s2p( 3 P o 1) 1s 2 ( 1 S 0 ) (Intercombination) z(m1) : 1s3s( 3 S 1 ) 1s 2 ( 1 S 0 ) (Forbidden) NOTE: 1s-2p are the K α transitions 32

33 2 & 3. PHOTOIONIZATION (PI) & RECOMBI- NATION (RC): Examples: Photoionization - in absorption spectra Recombination - in emission spectra 33

34 2 & 3. PHOTOIONIZATION (PI) & RECOMBI- NATION (RC): Inverse Processes i) Direct Photoionization (PI) & Radiative Recombination (RR) (1-step Bound-Continuum transitions): X +Z + hν X +Z+1 + ɛ ii) Autoionization (AI) & Dielectronic recombination (DR) (2-steps with intermediate autoionizing state): More exactly, X +Z + hν (X +Z 1 ) X +Z+1 + ɛ e + X +Z (X +Z 1 ) { e + X +Z AI X +Z 1 + hν DR The doubly excited state autoionizing state introduces resonances i & ii Energetic electron leaves/combines the atom Photoionization Cross Sections (σ P I ) Recombination Cross Sections (σ RC ) & Rate Coefficients (α RC ) - Show features with energy 34

35 Fig. Photoionization, Autoionization, Recombination: e - Autoionization e - Radiation-less decay X +n capture e - excitation e - AI X +n DR Recombined electron e - h Emission e - X +n Incoming electron energy - excites the core and binds the electron to an higher excited leve - short lived Quasi-bound, doubly excited state (AI) outer electron goes free when core drops down core drops down with emssion of a photon -outer electron becomes bound (DR) 35

36 ELECTRON-ION COLLISIONS: 1. Electron-impact excitation (EIE): e + X +Z e + X +Z e + X +Z + hν - Light is emitted as the excitation decays - show most common lines in astrophysical spectra - diagnostic forbidden lines - Scattered electron shows features with energy Can go through an autoionizing state Atomic quantity: Collision Strength (Ω) Fig. Excitation by electron impact: 36

37 2. Electron-impact Ionization (EII, important, but may not emit photon): e + X +Z e + e + X +Z+1 Atomic parameters: Ionization cross section (σ EII ) Ionizatin strength (S EII ) (often available experimentally) 37

38 THEORY ( Atomic Astrophysics and Spectroscopy, (AAS) Pradhan and Nahar, Cambridge 2011) Hamiltonian: For a multi-electron system, the relativistic Breit-Pauli Hamiltonian is: 1 2 H BP = H NR + H mass + H Dar + H so + N [g ij (so + so ) + g ij (ss ) + g ij (css ) + g ij (d) + g ij i j where the non-relativistic Hamiltonian is N H NR = 2 i 2Z N 2 + (6) r i i=1 the Breit interaction is j>i H B = [g ij (so + so ) + g ij (ss )] (7) i>j and one-body correction terms are H mass = α2 4 p 4 i, H Dar = α2 4 i ( ) 2 Z ri, H so = i 38 r ij

39 Ze 2 h 2 2m 2 c 2 r 3L.S Wave functions and energies are obtained solving: HΨ = EΨ E < 0 Bound (e+ion) states Ψ B E 0 Continuum states Ψ F 39

40 Transition Matrix elements with a Photon Dipole operator: D = i r i: < Ψ B D Ψ B > Photo-excitation & Deexcitation < Ψ B D Ψ F > Photoionization Matrix element is reduced to generalized line strength N+1 2 S = Ψ f r j Ψ i (8) j=1 Oscillator Strength, Photoionization, Recombination all are related to S Evaluation of the transition matrix element with a radiation field is non-trivial because of the exponential operator e ik.r describing the field e ik.r = 1 + ik.r + [ik.r] 2 /2! +..., (9) Various terms in the exponential expansion corresponds to various multipole transitions. The first term gives the electric dipole transitions E1, 2nd term E2 and M1,... The slection rules arise from the angular algebra of the transition matrices. 40

41 The exponential factor e ik.r in the transition matrix element gives the spatial dependence of the incident wave. An electron at a distance from the nucleus is of the order of Bohr radius, 10 8 cm. The magnitude of the wave vector k is 2π/λ. For visible light k is of the order of 10 5 cm 1 the wavelength is much larger than the size of an atom Retaining only the term unity in the first expansion, leads to the electric dipole approximation: e ik.r 1. f-, S, A-VALUES FOR VARIOUS TRANSITIONS Allowed electric dipole (E1) transtitions The oscillator strength (f ij ) and radiative decay rate (A ji ) for the bound-bound transition are [ ] f ij = ] S, A ji (sec 1 ) = 3g i [ Eji E3 ji 3g j S (10) i) Same spin multipliciy dipole allowed ( j=0,±1, L = 0, ±1, ±2, S = 0, parity π changes) ii) Intercombination ( j=0,±1, L = 0, ±1, ±2, S 0, π changes) Forbidden transitions i) Electric quadrupole (E2) transitions ( J = 0,±1,±2, parity does not change) A E2 ji = E5 ij g j S E2 (i, j) s 1, (11) 41

42 ii) Magnetic dipole (M1) transitions ( J = 0,±1, parity does not change) A M1 ji = E3 ij g j S M1 (i, j) s 1, (12) iii) Electric octupole (E3) transitions ( J= ±2, ±3, parity changes) A E3 ji = E7 ij g j S E3 (i, j) s 1, (13) iv) Magnetic quadrupole (M2) transitions ( J = ±2, parity changes) A M2 ji = s 1E5 ij g j S M2 (i, j). (14) LIEFTIME: The lifetime of a level can be obtained from the A-values, τ k (s) = 1 i A ki(s 1 ). (15) 42

43 Photoionization & Recombination cross section: Photoionization cross section is given by σ PI (Kα, ν) = 4π2 a 2 oαe ij ) S (16) 3 g k In central-field approximation, the photoionization cross section in length form is given by, [ σ L = 4π2 αω ij 1 ] l i R n j,l i +1 2 n 3 2l i + 1 i l i + (li + 1) R n j,l i 1 n i l i ]. (17) A general expression for the photoionization cross section of level n of a hydrogenic system is (Seaton 1959) ( σ n = 64παa2 o 3 ωn ) 3 ng(ω, n, l, Z), (18) 3 ω Z 2 where ω > ω n = α2 mc 2 Z 2 2, (19) 2 h n and g(ω, n, l, Z) is the bound-free Gaunt factor which is approximately unity at near-threshold energies. A more detailed expression for photoionization of a state nl is σ nl = 512π7 m 10 e 3 g(ω, n, l, Z)Z 4 (20) 3ch 6 n 5 ω 3 The photoionization cross section can be approximated by using hydrogenic wavefunctions that yield 43

44 the Kramer s formula, σ PI = 8π 1 (21) cn 5 ω 3 This is sometimes used to extrapolate photoionization cross sections in the high-energy region, where other features have diminished contribution with a relatively smooth background. However, it is not accurate. Since the spin does not change in a dipole transiton, and S i = S j, the angular algebra remains the same for both the bound-bound and bound-free transitions. So we have the same selection rules for photoionization, L = L j L i = 0, ±1, M = M Lj M Li = 0, ±1, l = l j l i = ±1. The length photoionization cross section is, σ L = π2 c 2 n ω ω 2 ijt ij = 4π2 α 3 ω ij l j =l i ±1 (2L j + 1) (l i + l j + 1) 2 L j (22) W 2 (l i L i l j L j ; L 1 1) < n j l j r n i l i > 2 (23) With α = e 2 /( hc) the cross section is in units of a 2 o and the constant in the equation is 4π2 a 2 oα 3 = Megabarns, 44

45 abbreviated as Mb = cm 2 The radial integral < n j l j r n i l i >= 0 R nj l j rr ni l i r 2 dr, (24) involves an exponentially decaying initial bound state wavefunction, and an oscillating continuum freeelectron wavefunction as a plane wave. The bound state wavefunctions are normalized to < i i > 2 =1, and free state wavefunctions are normalized per unit energy Recombination cross section, σ RC, from principle of detailed balance: g i h 2 ω 2 σ RC = σ PI g j 4π 2 m 2 c 2 v2. (25) The recombination rate coefficient: α RC (T) = 0 vf(v)σ RC dv, (26) f(v, T) = 4 π ( m 2kT )3/2 v 2 e mv2 2kT = Maxwellian velocity distribution function Recombination Rate Coefficient in terms of photoelectron energy is α RC (E) = vσ RC Total α RC Contributions from infinite number of recombined states 45

46 Radiative & Dielectronic Recombinations are inseparable in nature UNIFIED TREATMENT FOR ELECTRON-ION RECOMBINATION (Nahar & Pradhan, 1992, 1994, Zhang, Nahar, Pradhan, 1999) EXISTING METHODS: α RC = α RR + α DR UNIFIED METHOD: Total α RC - subsumes RR and DR including interference effects Considers infinite number of recombined states: i) Group (A) Low-n Bound States (n n o 10): σ RC are obtained from σ P I which includes autoionizing resonances. Hence subsumes both RR & DR ii) Group (B) High-n States (n o n ): These densely packed highly excited states, dominated by DR process, are treated through DR theory (Bell & Seaton 1985, Nahar & Pradhan 1994) in close-coupling approximation Advantages of the Unified method : Self-consistent treatment for photoionization & recombination - use of same wavefunction Provides level-specific recombination rate coefficients & photoionization cross sections for a large number of bound levels 46

47 Ionization Fractions in Astrophysical Plasma Equilibria: Unified Treatment of Photoionization & Electron-Ion Recombination i) Plasma in Coronal Equilibrium: Electron-Impact Ionization (EII) is balanced by Electron Ion Recombination: N(z 1)S EII (z 1) = N(z)α RC (z) S EII (z 1) = total EII rate coefficient, α RC = total recombination rate coefficient ii) Plasma in Photoionization Equilibrium: With a radiation flux J ν, photoionization is balanced by electron-ion recombination: N(z) ν 0 4πJ ν hν σ PI(z, ν)dν = N e N(z + 1)α RC (z, T e ) ν o = Ionization potential of the ion, σ PI = Ground state photoionization cross sections Accurate ionization fractions, N i /N, requires Unified treatment of total (RR+DR) rate coefficient Self-consistency between these inverse processes 47

48 Ionization Fractions at Coronal Equibrium: O-ions (Nahar 1998) 1 O ions I II III IV 0 IX -1 log 10 [N(z)/N T ] -2-3 VIII -4 VII -5 VI -6 V log 10 T (K) Iron Ionization Fractions at Photoionization Equilibrium in a typical PNe: Fe IV dominates in most of the nebula, Fe V fraction is reduced by half (Nahar & Bautista 1995) 1 Fe II.8 Fe VI Fe IV Fe III Ionic fraction.6.4 Fe V Radius [cm]

49 EIE: Scattering matrix S SLπ (i, j) is obtained from the wave function phase shift. The EIE collision strength is 1 2 SLπ Ω(S i L i S j L j ) = l i l j (2S + 1)(2L + 1) S SLπ (S i L i l i S j L j l j ) 2 The effective collision strength Υ(T) is the Maxwellian averaged collision strength: Υ(T) = 0 Ω ij (ɛ j )e ɛ j/kt d(ɛ j /kt) (27) The excitation rate coefficient q ij (T) is q ij (T) = g i T 1/2 e E ij/kt Υ(T) cm 3 s 1 (28) E ij = E j E i and E i < E j in Rydbergs 49

50 Monochromatic Opacities κ ν of Fe II: (Nahar & Pradhan 1993) TOP: Complex structure in monochromatic opacity of Fe II at the Z-bump temperature and density Bottom: The high opacity in region indicates the absorption by Fe I and Fe II in the solar blackbody radiation. 8 Monochromatic opacity κ(fe II) Log T = 4.1, Nε = Log κ (a o 2 ) Log λ (A) 3 F (10 6 erg cm 2 s 1 A 1 ) λ λ (A)

51 PHOTOIONIZATION An atom/ion with more than 1 electron will have resonances in σ P I Earlier calculations for photoionization cross sections under the Opacity Project considered low-lying resonances Core excitations to higher states during photoionization are thought to be weaker and behave hydrogenic Fine structure introduce new features that can not be obtained in LS coupling New calculations under the Iron Project are showing new and dominating features not studied before - these are expected to resolve some longstanding problems s H I (ns 2 S) + hν -> H II + e (Nahar 1996) s 3s 4s 5s 10-6 σ PI (Mb) He I (1s 2 1 S) + hν -> He II + e (Nahar 2010) n=2 threshold of the core Photon Energy (Ry)

52 Physics Experiment: Photoionization cross sections of N IV: TOP: measured at synchrotron facility BESSY II by Simon et al (2010) BOTTOM: Comparison with NORAD-Atomic-Data (Nahar & Praadhan 1997, blue). MCDF (Orange drop lines) 52

53 Photometric Image: Orion Constellation Red gaseaous nebula in the background - Orion Nebula (reddish patch) below the belt of 3 stars (on the sword) Below: Imaginary figure - Orion the Hunter 53

54 Determination of Abundance of Elements Orion Nebula - Birthplace of Stars [Composed by images from Spitzer & Hubble] Orion Nebula 1500 Lyr away, closest cosmic cloud to us Center bright & yellow gas - illuminated ultraviolet (UV) radiation Images: Spitzer - Infrared (red & orange) C rich molecules - hydrocarbons, Hubble - optical & UV (swirl green) of H, S Small dots - infant stars; over 1000 young stars Detected lines of O II, O III 54

55 Determination of Abundance of Elements Longstanding problems on determination of abudances of elements - lack of accurate atomic data One is Oxygen abundance in Nebular plasmas Ex: Planetary Nebula (PNe) K 4-55 (below) Planetary nebula (PNe): last stage of a star Has condensed central star: very high T 100,000 K (>> T 40,000 K of a typical star) Envelope: thin gas radiatively ejected & illuminated by radiation of the central star: red (N), blue (O) Common lines of low ionization states: O III and O II - 55 low density and low T

56 Determination of OXYGEN Abundance: Collisionally Excited Lines (CEL): e + O III O III* O III + hν Recombination lines (REL): e + O III O II + hν The intensity of a CEL of ion X I due to transition between a and b (Atomic Astrophysics and Spectroscopy, Pradhan & Nahar, 2011) I ba (X i, λ ba ) = hν 4π n en ion q ba (29) q ba - EIE rate coeffcient in cm 3 /sec. Hence population of a level b can be written as N(b) = n ion q ba = 1 n e 4π hν I ba (30) The abundance, n(x)/n(h) of the element, X, with respect to H, can be obtained from the intensity I I(X i, λ ba ) = hν [ ] 4π A N(b) n(x ba i ) n(x) j N n(h) j(x i ) n(x) n(h) A ba - radiative decay rate, sum over N j - total populations of all excited levels, N(b)/ j N j(x i ) - population fraction, (obtained 56

57 from a radiative -collisional model), n(x i) n(x) (come from photoionization model code) - ionization fraction Similarly abundance can obtaioned from the intensity of a REL from emissivity ɛ(λ pj ) = N e N(X + )α eff (λ pj )hν pj [arg cm 1 s 1 ] intensity, cascade matrix, and formula similar to that of CEL but for REL Problem: N e N(O III)q EIE < α R (T)N e N(O II) ONE SOLUTION: Recombination rate for (e + O III) O II - higher at low T 57

58 Close-Coupling (CC) Approximation & the R-matrix method CC approximation: Ion treated as a (N+1) electron system: - a target or the ion core of N electrons and - an additional interating (N+1)th electron: Total wavefunction expansion is: N Ψ E (e + ion) = A χ i (ion)θ i + c j Φ j (e + ion) i j χ i target ion or core wavefunction θ i interacting electron wavefunction (continuum or bound) Φ j correlation functions of (e+ion) The complex resonant structures in the atomic processes are included through channel couplings. Substitution of Ψ E (e+ion) in HΨ E = EΨ E results in a set of coupled euqations Coupled equations are solved by R-matrix method 58

59 Close-Coupling Approximation The wavefunction is an expansion: N Ψ E (e + ion) = A χ i (ion)θ i + i j c j Φ j (e + ion) χ i core ion wavefunction, Φ j correlation functions θ i interacting electron wavefunction (continuum or bound) Resonant Structures - manifested through channel couplings Table (Example of Ψ E ): Levels and energies (E t ) of the core O III in wave function expansion of O II; SS - calculated energies and NIST - observed energies Level J t E t (Ry) E t (Ry) NIST SS 1 1s 2 2s 2 2p 2 ( 3 P ) s 2 2s 2 2p 2 ( 3 P ) s 2 2s 2 2p 2 ( 3 P ) s 2 2s 2 2p 2 ( 1 D) s 2 2s 2 2p 2 ( 1 S) s 2 2s2p 3 ( 5 S o ) s 2 2s2p 3 ( 3 D o ) s 2 2s2p 3 ( 3 D o ) s 2 2s2p 3 ( 3 D o ) s 2 2s2p 3 ( 3 P o ) s 2 2s2p 3 ( 3 P o ) s 2 2s2p 3 ( 3 P o ) s 2 2s2p 3 ( 1 D o ) s 2 2s2p 3 ( 3 S o ) s 2 2s2p 3 ( 1 P o ) s 2 2s2p3s( 3 P o ) s 2 2s2p3s( 3 P o ) s 2 2s2p3s( 3 P o ) s 2 2s2p3s( 1 P o )

60 R-matrix Method Substitution of Ψ E (e+ion) in HΨ E = EΨ E a set of coupled equations - solved by the R-matrix method Divide the space in two regions, the inner and the outer regions, of a sphere of radius r a with the ion at the center. r a is large enough for to include the electron-electron interaction potential. Wavefunction at r > r a is Coulombic due to perturbation In the inner region, the radial part F i (r)/r of the outer electron wave function (θ) is expanded in terms of a basis set, called the R-matrix basis, F i = a k u k [ ] d 2 l(l + 1) + V(r) + ɛ dr2 r 2 lk u lk + λ nlk P nl (r) = 0. n (31) and is made continuous at r a with the Coulomb functions outside r a 60

61 R-Matrix Codes For Large-Scale Atomic Calculations at the Ohio Supercomputer Center VARIOUS COMPUTATIONAL STAGES R-matrix calculations cen proceed in 3 branches - 1) LS coupling & relativistic Breit-Pauli, 2) LS coupling R-matrix II for Large configuration interaction, 3) DARC for Full Dirac relativistic Results - 1) Energy Levels, 2) Oscillator Strengths, 3) Photoionization Cross sections, 4) Recombination Rate Coefficients, 5) Collision Strengths; - Astrophysical Models THE R MATRIX CODES AT OSU ATOMIC STRUCTURE: CIV3 OR SUPERSTRUCTURE GRASP R MATRIX R MATRIX II DIRAC R MATRIX STG1 RAD DSTG1 *FULL STG2 ANG DSTG2 BREIT PAULI* RECUPD LS HAM DSTGH STGH DIG H DSTG3 D H STGB *STGF(J)* PFARM *DSTGFR* DSTGF DSTG4 B F *ELEVID* /*PRCBPID* STGBB STGBF / *STGBFRD* P *STGRC* ENERGY OSCILLATOR PHOTOIONIZATION RECOMBINATION COLLISION LEVELS STRENGTHS CROSS SECTIONS CROSS SECTIONS STRENGTHS ASTROPHYSICAL AND PLASMA SPECTRAL MODELS AND OPACITIES 61

62 ELECTRON IMPACT EXCITATIONS (EIE) Fig Collision strengths Ω (EIE) of O III: 2p 2 ( 3 P 0 3 P 1 ) (88µm) and 2p 2 ( 3 P 1 3 P 2 ) (52µm) (IR) (Palay et al 2012) The near threshold resonances are seen for the first time Collision Strength Collision Strength Ω( 3 P 0 3 P 1 ) Energy (Ryd) Ω( 3 P 1 3 P 2 ) Energy (Ryd) a) b) Fig Effective Ω(T) of 3 optical lines; Solid: BPRM (present), Dashed: R-matrix(LS) (Aggarwal & Keenan) Differences to affect T and density diagnostics Effective Collision Strength υ(t) Å 4959 Å 4363 Å Log T(K) 62

63 LINE RATIO: DENSITY & T DIAGNOSTICS Intensity ratio of two observed lines can be compared to the calculated curves for density (ρ) & T diagnostics Fig Significant FS effect on ρ diagnostics, ,000 K a) 10,000 K b) 10,000 K 52/88 µm ,500 K 52/88 µm ,000 K 500 K 315 K K Log N e (cm 3 ) Log N e (cm 3 ) Blended line ratio with T at N e = 10 3 cm 3. The diagnostic indicates considerable rise in low T region 15 Log[( )Å/4363Å] Log T (K) 63

64 Fine Structure Effects on Low Z ion: O II (Nahar et al. 2010) Previous study of σ PI for O II in LS coupling R-matrix (Nahar 2004) showed good agreement with experiment (ALS: Covington et al. 2001) However, problem remained with O II abundance at low T astrophysical plasmas σ P I in fine structure (FS) is very similar to that in LS coupling except the resonant structure at the threshold The new low energy features in σ PI (FS) should narrow down the discrepancy 20 Photoionization: O II + hν -> O III + e 15 a) 4 S o :LS 10 5 σ PI (Mb) b) 4 S o 3/2 :FS Photon 64 Energy (Ry)

65 GROUND STATE PHOTOIONIZATION OF O II: Photoionization of Ground state 4 S o of O II in small energy region near threshold Panel a) (Top) Photoionization in LS coupling - no structure in the region (Nahar 2004) Panel b) Total σ P I, of the ground level 4 S3/2 o : i) filled with resonant structures, ii) background shifts (Nahar et al. 2010) Panels c,d,e) Partial photoionization into fine structure components 2p 2 3 P 0,1,2 of O iii, arrows ionization thresholds. Photoionization Cross sections of O II 10 2 O II (2s 2 2p S o ) + hν -> O III + e a) Ground State Total: LS b) Ground State Total: Full Breit-Pauli C=2s 2 2p 4 3 P 0 (partial) (c) σ PI (Mb) C=2s 2 2p 4 3 P (partial) (d) C=2s 2 2p 4 3 P (partial) (e) Photon Energy (Ry) 65

66 PHOTOIONIZATION OF O II: PEC RESONANCES PEC (Photo- Excitation-of-Core) resonances appear photoionization of single electron valence excited levels Figure shows that PEC resonances are strong and enhance the background cross sections by orders of magnitude These PEC resonances will affect photoionization and recombination rates of plasmas Photoionization of O II: PEC Resonances 10 2 O II(2s 2 2p 2 3d 4 D 1/2 ) + hν -> O III + e σ PI (Mb) Photon Energy (Ry) 66

67 New RECOMBINATION CROSS SECTION AT LOW T: O II (Nahar et al. 2010) Recombination cross section σ RC show many resonances at very low energy Recombination rate from these will increase the recombination rate Higher recombination rate will decrease the oxygen abundance and hence will resolve or narrow down the existing gap in nebular problem Recombination Cross Sections of O II 10 3 e + O III -> O II 10 2 a) Ground State: 2s 2 2p 5 ( 4 S o 3/2 ) σ RC (Mb) b) Sum of State: 2s 2 2p 5 ( 4 S o, 2 D o, 2 P) Photoelectron Energy (ev)

68 O II: Ground State RECOMBINATION (RR+DR): Fig: Recombination rate coefficients (RRC) α RC (T) for the ground state Fine structure effects have increased RRC considerably at low temperatre This indicates lower abundance of oxygen Recombination Rate Coefficients of O II Ground State Recombination: O III + e -> O II α R (nls) (cm 3 s -1 ) s 2 2p 3 ( 4 S o 3/3 ): FS s 2 2p 3 ( 4 S o 3/3 ): LS T(K) 68

69 O II: TOTAL RECOMBINATION (RR+DR): Fig: Total recombination rate coefficients (RRC) α RC (T) Fine structure effects have increased RRC considerably, by 2.8 times around 10 K and 2.2 at 100 K, The high temperature DR peak has been reduced some. The reason could be some radiation damping effect included in the present work. The resolution of the narrow fine structure resonances could also affect the decrease Total Recombination Rate Coefficients O II O III + e -> O II α R (T) (cm 3 sec -1 ) a) Relativistic (present) b) LS coupling (Nahar 1999) T (K) 69

70 PLASMA OPACITY Opacity is a fundamental quantity for plasmas - it determines radiation transport in plasmas Opacity is the resultant effect of repeated absorption and emission of the propagating radiation by the constituent plasma elements. Microscopically opacity depends on two radiative processes: i) photo-excitation (bound-bound transition) & ii) photoionization (bound-free transition) The total κ(ν) is obtained from summed contributions of all possible transitions from all ionization stages of all elements in the source. Calculation of accurate parameters for such a large number of transitions has been the main problem for obtaining accurate opacities. 70

71 The OPACITY Project & The IRON Project AIM: Accurate Study of Atoms & Ions, Applications to Astronomy International Collaborations: France, Germany, U.K., U.S., Venezuela, Canada, Belgium Earlier opacities were incorrect by factors of 2 to 5 inaccurate stellar models initiation of the Opacity Project in 1981 THE OPACITY PROJECT - OP ( ): Studied radiative atomic processes for (E, f, σ PI ) Elements: H to Fe Calculated opacities of astrophysical plasmas THE IRON PROJECT - IP (1993 -): collisional & radiative processes of Fe & Fe peak elements RMAX: Under IP, study X-ray atomic astrophysics Atomic & Opacity Databases (from OP & IP) TOPbase (OP) at CDS: TIPbase (IP) at CDS: OPserver for opacities at the OSC: Latest radiative data at NORAD-Atomic-Data at OSU: nahar/ nahar radiativeatomicdata/index.html 71

72 ACHIEVEMENTS OF THE OP & THE IP The first detailed study of the atomic processes for most of the atoms and ions Found new features in photoionization cross sections OP opacities agreed with those computed under the OPAL, and both solved the outstanding problem on pulsations of cepheids Results from the OP and the IP continue to solve many outstanding problems, e.g., spectral analysis of blackhole environment, abundances of elements, opacities in astrophysical plasmas, dark matter HOWEVER, these are not complete and sufficiently accurate enough to solve all astrophysical problems 72

73 Observation & Modeling: Emission spectra of Fe I - III in active galaxy 1 Zwicky 1 (Sigut, Pradhan, Nahar 2004) Relative Flux FeII with Lyα Fe III Obs 10 m3 10 p0 10 p3 20 m3 20 p0 20 p3 40 m3 40 p0 40 p3 0.5 Relative Flux FeII without Lyα Fe III Obs 10 m3 10 p0 10 p3 20 m3 20 p0 20 p3 40 m3 40 p0 40 p Wavelength (Angstroms) Blue - Observation; Curves - Various Models with 1000 energy levels, millions of transitions With (top) and without (bottom) Lyman-alpha fluorescent excitation of Fe II by recombining H- atoms. The models reproduce many of the observed features, but discrepancies indicate need for more accurate calculations. 73

74 SOLAR OPACITY (Seaton et al 1993) 4 Log (R) Solar Opacity Log κ R (cm 2 /g) Log T(K) The Rosseland mean opacity κ R in several temeprature-density regimes throughout the solar interior characterized by the parameter R = ρ(g/cc)/t 3 6, T 6 = T 10 6 For the sun, -6 R -1 The bumps and kinks in the curves represent higher opacities due to excitation/ionization of different atomic species at those temperatures (H-, He-, Z-, and inner-shell bumps) 74

75 SOLAR OPACITY & ABUNDANCES At the convection zone boundary, R CZ, the temperature T e 193 ev (2 MK), density n e /cm 3 (HED - High Energy Density) HED condition important elements: O, Ne, especially Fe (Fe XVII-XIX) Recent determination from measurements and 3D hydro NLTE models: 30-40% lower abundances of C, N, O, Ne, Ar than the standard abundances Optical depth in the sun depends on the interior opacity (κ) through elemental abundances The opacity can determine R CZ 75

76 DISCREPANCY IN SOLAR RADIAIVE AND CONVECTION ZONES BOUNDARY (R CZ ) The measured boundary, from helioseismology, of R CZ is The calculated R CZ is large 76

77 77

78 Determination of OPACITY: RADIATIVE ATOMIC PROCESSES 1. Photoexcitation - Photon absorption for a boundbound transition X +Z + hν X +Z Oscillator Strength (f ij ) Monochromatic opacity (κ ν ) depends on f ij κ ν (i j) = πe2 mc N if ij φ ν (32) N i = ion density in state i, φ ν = profile factor κ inclues 100M transitions of mid-z elements 2. Photoionization - Photon absorption for a boundfree transition: Direct - X +Z + hν X +Z+1 + e 3. Autoionization (AI) in photoionization process : { e + X e + X +Z (X +Z 1 ) +Z AI X +Z 1 + hν DR Doubly excited autoionizing state resonance Photoionization Cross Sections (σ P I ) 78

79 κ ν depends on σ P I κ ν = N i σ PI(ν) (33) κ ν depends also on processes Inverse Bremstrahlung free-free Scattering: hν + [X e(ɛ)] X e(ɛ ), (34) Cross section - from the elastic scattering matrix elements for electron impact excitation. An approximate expression for the free-free opacity is κ ff ν (1, 2) = N e N i g ff Z 2 where g ff is a Gaunt factor T 1/2 ν 3 (35) Photon-Electron scattering: a) Thomson scattering when the electron is free κ(sc) = N e σ Th = N e 8πe 4 3m 2 c 4 = cm 2 /g (36) b) Rayleigh scattering when the electron is bound κ R ν = n i σ R ν n i f t σ Th ( ν ν I ) 4 (37) hν I = binding energy, f t = total oscillator strength associated with the bound electron. The total monochromatic opacity is then given by κ ν = κ ν (bb) + κ ν (bf) + κ ν (ff) + κ ν (sc) (38) 79

80 Mean Opacity: The equation of state (EOS) Requires ionization fractions and level populations of each ion of an element in levels with nonnegligible occupation probability Saha equation - number density for ionization state n S (X i ) of excited ion X i at thermal equilibrium as n S (X i ) = g i 2g s h 3 (2πm e kt) 3 2 exp ( E ) gi n e n(x + ) (39) kt In LTE the level populations are given by the Boltzmann equation, N i = g i e hν/kt (40) N j g j The combination of the Saha-Boltzmann equations specify the Equation-of-State (EOS), in LTE N ij = N jg ij e E ij/kt U j (41) g ij = statistical factor of level i of ionization state j, the partition function is U j = g ij e ( E ij/kt) (42) i There are several ways to modify the Saha-Boltzmann equations to incorporate the effect of plasma interactions into the EOS. The modified density function 80

81 is then N ij = N jg ij w ij e E ij/kt U j, (43) where the partition function is U j = i g ij w ij e ( E ij/kt) (44) Rosseland mean κ R (T, ρ): Harmonic mean opacity averaged over the Planck function, ρ is the mass density (g/cc), 1 κ R = 0 1 κ ν g(u)du 0 g(u)du, where g(u) is the Planck weighting function g(u) = 15 u 4 e u hν 4π 4 (1 e u ) 2, u = kt g(u), for an astrophysical state is calculated with different chemical compositions H (X), He (Y) and metals (Z), such that X + Y + Z = 1 Solar abundaces: X=0.7, Y=0.28, Z=

82 PHOTOIONIZATION CROSS SECTION: Fe XVII Top: Ground level: n=2 resonances are important Bottom: Excited level: n=3 resonances are important Arrows point energy limits of n=2 & 3 core states Photoionization Cross sections of Fe XVII 10 2 Fe XVII + hν -> Fe XVIII + e 10 1 a) Ground state: 2s 2 2p 6 1 S n=2 n=3 σ PI (Mb) b) Excited Level: 2s 2 2p 5 3d 3 P o n=2 n= Photon Energy (Ry) 82

83 COMPARISON OF PHOTOIONIZATION CROSS SECTIONS Photoionization cross sections of Fe XVII: (a,b) level 2p 5 3p 1 P & (c,d) level 2p 5 3d( 1 D o ) Present σ P I (Nahar et al 2010) is much more highly resolved than those given by the Opacity Project (OP). Without inclusion of n=3 core states, σ P I is considerably underestimated 10 2 Fe XVII + hν -> Fe XVIII + e 10 2 Fe XVII + hν -> Fe XVIII + e a) 2p 5 3p( 1 P) OP c) 2p 5 3d( 1 D o ) OP σ PI (Mb) b) 2p 5 3p( 1 P) d) 2p 5 3d( 1 D o ) 10 1 Present 10 1 Present n=2 n= n=2 n= Photon Energy (Ry)

84 MONOCHROMATIC OPACITY OF Fe XVII (Nahar et al. 2011) Monochromatic opacity κ of Fe XVII at temperature T = K and electron density N e = cm 3, corresponding to the base of the solar convection zone where the ion is the largest contributor to opacity. 5 BPRM With Resonances Log κ (cm 2 /g) 1 5 OP Photon Energy E(KeV) 84

85 85

86 and Arizona, Germany, Italy Large Binocular Telescope (LBT) Largest Telescope: 8.4m Mirrors (11.8m), NIR-Optica Galaxy: M101 Galaxy: M51 Galaxy: NGC